Solutions Manual to Accompany Linear Algebra

STUDENT MANUAL 1
1 SYSTEMS OF LINEAR EQUATIONS 3
1.1 The Vector Space of m × n Matrices / 3
1.1.2 Applications to Graph Theory I / 7
1.2 Systems / 8
1.2.2 Applications to Circuit Theory / 11
1.3 Gaussian Elimination / 13
1.3.2 Applications to Traffic Flow / 18
1.4 Column Space and Nullspace / 19
2 LINEAR INDEPENDENCE AND DIMENSION 26
2.1 The Test for Linear Independence / 26
2.2 Dimension / 33
2.2.2 Applications to Differential Equations / 37
2.3 Row Space and the Rank-Nullity Theorem / 38
3 LINEAR TRANSFORMATIONS 43
3.1 The Linearity Properties / 43
3.2 Matrix Multiplication (Composition) / 49
3.2.2 Applications to Graph Theory II / 55
3.3 Inverses / 55
3.3.2 Applications to Economics / 60
3.4 The LU Factorization / 61
3.5 The Matrix of a Linear Transformation / 62
4 DETERMINANTS 67
4.1 Definition of the Determinant / 67
4.2 Reduction and Determinants / 69
4.2.1 Volume / 72
4.3 A Formula for Inverses / 74
5 EIGENVECTORS AND EIGENVALUES 76
5.1 Eigenvectors / 76
5.1.2 Application to Markov Processes / 79
5.2 Diagonalization / 80
5.2.1 Application to Systems of Differential Equations / 82
5.3 Complex Eigenvectors / 83
6 ORTHOGONALITY 85
6.1 The Scalar Product in n / 85
6.2 Projections: The Gram-Schmidt Process / 87
6.3 Fourier Series: Scalar Product Spaces / 89
6.4 Orthogonal Matrices / 92
6.5 Least Squares / 93
6.6 Quadratic Forms: Orthogonal Diagonalization / 94
6.7 The Singular Value Decomposition (SVD) / 97
6.8 Hermitian Symmetric and Unitary Matrices / 98
7 GENERALIZED EIGENVECTORS 100
7.1 Generalized Eigenvectors / 100
7.2 Chain Bases / 104
8 NUMERICAL TECHNIQUES 107
8.1 Condition Number / 107
8.2 Computing Eigenvalues / 108